CS23: Stochastic Processes Under Constraints

date: 7/17/2025, time: 14:00-15:30, room: IM 601

Organizer: Dominic T. Schickentanz (Paderborn University, Germany)

Chair: Dominic T. Schickentanz (Paderborn University, Germany)

Partially-homogeneous reflected random walk on the quadrant

Andrew Wade (Durham University, UK)

We consider a random walk on the first quadrant of the square lattice, whose increment law is, roughly speaking, homogeneous along a finite number of half-lines near each of the two boundaries, and hence essentially specified by finitely-many transition laws near each boundary, together with an interior transition law that applies at sufficient distance from both boundaries. Under mild assumptions, in the (most subtle) setting in which the mean drift in the interior is zero, we classify recurrence and transience and provide power-law bounds on tails of passage times; the classification depends on the interior covariance matrix, the (finitely many) drifts near the boundaries, and stationary distributions derived from two one-dimensional Markov chains associated to each of the two boundaries. As an application, we consider reflected random walks related to multidimensional variants of the Lindley process, for which no previous quantitative results on passage-times appear to be known.

Based on joint work with Conrado da Costa and Mikhail Menshikov (Durham University).

Persistence probabilities of spherical fractional Brownian motion

Max Helmer (Technische Universität Darmstadt)

We consider spherical fractional Brownian motion \((S_H(\eta))_{\eta\in\mathbb{S}_{d-1}}\), which is obtained by taking fractional Brownian motion indexed by the (multi-dimensional) sphere \(\mathbb{S}_{d-1}\), and calculate its persistence exponent. Persistence in this context is the study of the decay of the probability \[\mathbb{P}\left( \sup_{\eta \in \mathbb{S}_{d-1}} S_H(\eta) \leq \varepsilon \right)\] when the barrier \(\varepsilon \searrow 0\) becomes more and more restrictive. Our main result shows that the persistence probability of spherical fractional Brownian motion has the same order of polynomial decay as its Euclidean counterpart. This is joint work with Frank Aurzada (TU Darmstadt).

Brownian Motion Subject to Time-Inhomogeneous Additive Penalizations

Dominic T. Schickentanz (Paderborn University)

Consider a Brownian motion \(B=(B_t)_{t \ge 0}\), started in \(x \in \mathbb{R}\), as well as a positive random variable \(\xi\) independent of \(B\) and a measurable, locally bounded function \(u: \mathbb{R}\times [0,\infty) \to~[0,\infty)\). Let \[\tau:= \inf\left\{T \ge 0: \int_0^T u(B_s,s) \mathrm{d}s \ge \xi\right\}\] be the first time the time-inhomogeneous additive Brownian functional associated with \(u\) reaches the threshold \(\xi\). We will analyze the asymptotic behavior of \(\mathbb{P}_x(\tau >T)\) as \(T \to \infty\) and, in particular, provide sufficient criteria for this probability to decay like a multiple of \(\frac{1}{\sqrt{T}}\). Subsequently, we will discuss the existence and long-term behavior of the associated conditioned process, i.e., of \(B\) conditioned on the rare event \[\{\tau=\infty\} = \left\{\int_0^t u(B_s,s) \mathrm{d}s <\xi \text{ for all } t \ge 0\right\}.\] Our framework, in particular, covers occupation times below a wide range of moving barriers. Further, it covers the case where \(u\) is a modified solution of the FKPP equation. This will be the key to upcoming results concerning branching Brownian motions with critically large maximum, a joint project with Bastien Mallein (Toulouse).